Journal of Scientific Research & Reports 3(11): 1430-1448, 2014; Article no. JSRR.2014.11.002 SCIENCEDOMAIN international www.sciencedomain.org Identifying Stiff Ordinary Differential Equations and Problem Solving Environments (PSEs) B. K. Aliyu1*, C. A. Osheku1, A. A. Funmilayo1 and J. I. Musa1 1Federal Ministry of Science and Technology (FMST), National Space Research and Development Agency (NASRDA), Centre For Space Transport and Propulsion (CSTP) Epe, Lagos State, Nigeria. Authors’ contributions This work was carried out in collaboration between all authors. Author BKA designed the study, carried out the simulations and wrote the first draft of the manuscript. Author CAO supervised the study and re-drafted the manuscript. Authors AAF and JIM verified all simulations results and literature search. All authors read and approved the final manuscript. Received 3rd March 2014 th Original Research Article Accepted 4 April 2014 Published 16th April 2014 ABSTRACT Stiff Ordinary Differential Equations (SODEs) are present in engineering, mathematics, and sciences. Identifying them for effective simulation (or prediction) and perhaps hardware implementation in aerospace control systems is imperative. This paper considers only linear Initial Value Problems (IVPs) and brings to light the fact that stiffness ratio or coefficient of a suspected stiff dynamic system can be elusive as regards the phenomenon of stiffness. Though, it gives the insight suggesting stiffness when the value is up to 1000 but is not necessarily so in all ODEs. Neither does a value less than 1000 imply non-stiffness. MATLAB/Simulink® and MAPLE® were selected as the Problem Solving Environment (PSE) largely due to the peculiar attribute of Model Based Software Engineering (MBSE) and analytical computational superiority of each PSEs, respectively. This creates the base for comparing results from both numerical and analytical standpoint. In Simulink, two methods of modelling ODEs are presented. Experimenting with all the variable-step solvers in MATLAB® ODE Suit for selected examples was carried out. Results point to the fact that stiffness coefficient of about 1000does not always suggest that an ODE is stiff nor does a value less than 1000 suggest non-stiff. Keywords: ODEs; stiffness ratio; MATLAB/Simulink®; MAPLE®; variable-step solvers. ____________________________________________________________________________________________ *Corresponding author: E-mail: [email protected]; Aliyu et al.; JSRR, Article no. JSRR.2014.11.002 1. INTRODUCTION An ordinary linear differential equation is known to model any linear system, relating the output response of the system to the input, whether electrical, mechanical, hydraulic, or thermal. The analogies between these several engineering disciplines have been developed over time [1]. ODE problem has been divided into stiff and non-stiff problems. In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proved difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution. Stiffness is a subtle, difficult and important concept in the numerical solution of ordinary differential equations. It depends on the differential equation, the initial conditions, and the numerical method. The best way to detect stiffness is to try one of the solvers intended for non-stiff systems. If it is unsatisfactory, the problem maybe stiff. If the problem is stiff, there are effective solvers available [2]. The phenomenon of stiffness is not precisely defined in literature. Some attempts at describing a stiff problem are: A differential equation of the form y’= f(t,y(t)) is said to be stiff if its exact solution y(t) includes a term that decays exponentially to zero as t increases, but whose derivatives are much greater in magnitude than the term itself. An example of such a term is e-λt, where λ is a large, positive constant, because its kth derivative is cke-λt. Because of the factor of ck, this derivative decays to zero much more slowly than e-λt as t increases. Because the error includes a term of this form, evaluated at a time less than t, the error can be quite large if h which is the step size is not chosen sufficiently small to offset this large derivative. Furthermore, the larger λ is, the smaller h must be to maintain accuracy. A problem is stiff if it contains widely varying time scales, i.e., some components of the solution decay much more rapidly than others, A problem is stiff if the step size is dictated by stability requirements rather than by accuracy requirements. A problem is stiff if explicit methods don’t work, or work only extremely slowly. A linear problem is stiff if all of its eigenvalues have negative real part, and the stiffness ratio (the ratio of the magnitudes of the real parts of the largest and smallest eigenvalues) is large. It is pertinent to note that in this research we are concerned with the computational aspect of the properties that define stiffness. If we weren't concerned with how many computational steps an algorithm takes to converge to a solution which implies more storage space on a microcontroller for control system hardware implementation, we wouldn't be concerned about stiffness. Non-stiff methods can solve stiff problems; they just take a long time to do it. Also, more computational steps translate into longer time for an algorithm to converge to a solution. Both issues are crux in control system design. 1431 Aliyu et al.; JSRR, Article no. JSRR.2014.11.002 2. STIFFNESS RATIO OR COEFFICIENT The following definitions are popular as regards a numerical factor attributed to the degree of stiffness: Definition 1.1 considering a higher order system defined as y Ay y t0 y 0 , (1) with an n× n matrix A having eigenvalues λi(i=1,…,n) which all have a negative real part. If these real parts differ considerably, the initial value problem is called stiff. A measure for the stiffness is the so called stiffness coefficient, expressed as maxi ( i ) S1 : (2) mini ( i ) Note that both λmax and λmin must be greater than zero for stiffness to occur. If the coefficients of the ODE (i.e., the matrix elements) comprise several orders of magnitude, S can reach values of ϑ(106) or even more. Definition 1.2 generally, a dimensionless index of stiffness for every system is given as S2 b a, (3) Where [a, b] is the integration interval of interest, Re i , and λi is the eigenvalue of the largest negative real part. Actually τ varies along the solution. While the intuitive meaning of stiff is clear to all specialists, much controversy is going on about its correct mathematical definition. The most pragmatic opinion is also historically the first one which states that Stiff equations are equations where certain implicit methods, in particular backward differentiation methods, perform much better than explicit ones. Stiff ODEs defeat the explicit Runge-Kutta and Adams-Bashforth methods. In fact, for such problems, the higher order methods perform even more poorly than the low order methods [3]. 2.1 Stiffness The common approach to understanding stiffness is motivated by the behaviour of fixed step size solutions for systems of linear ODEs with constant coefficients. The eigenvalues of the coefficient matrix completely characterize the solution of the system and they determine the behaviour of numerical methods applied to the system. Generally, stiffness is considered tobe important for systems in which all of the eigenvalues λi of the coefficient matrix have negative real parts, that is, to systems that are mathematically stable. When explicit methods such as those in ode45are used, stiffness imposes restrictions on the step size required to obtain a stable numerical approximation. It is easy to see how such restrictions arise by considering what happens when the Forward Euler method given in (4), 1432 Aliyu et al.; JSRR, Article no. JSRR.2014.11.002 with h as the step size and tn+1=tn+ h is a sequence of time at which the approximate solution yn+1is desired yn1 y n hf t, y n , (4) is applied to the linear test equation given in (5).A simple calculation shows that the numerical solution is given by (6) y t i y t , y 0 1. (5) n yn1 h i . (6) The numerical solution decays to 0 as tn increases only if |(1 + hλi) |<1.The complex values for which this is the case form the interior of a unit circle cantered at the point (-1,0). Since this restriction must hold for each of the eigenvalues, it imposes a severe restriction on the step size when Sis large. Any explicit method has a finite stability region determined by the above condition. By way of contrast, some implicit methods have infinite stability regions. A first example is the Backward Euler method given in (7) which generates the approximations given by (8). yn1 y n hf t, y n 1 . (7) 1 yn n . (8) 1 hi These approximations decay to 0 for points exterior to the unit circle centred at the point (1,0). In particular, the resulting stability region contains the entire left half-plane. If the ODE system is stable, no numerical stability restriction is thus placed on the step size which is then controlled by keeping local errors sufficiently small. The MATLAB stiff ODE solver ode15s implements this method and similar higher order ‘stiffly stable’ methods whose stability regions contain a significant portion of the left half-plane. Imagine that an explicit method such as ode45 is being used. Step sizes will be used for which hS is near the stability boundary for the method and for some step sizes hS will fall outside the stability region.